So,

I have quite freshly installed Debian 9 with Mate de.

Text-files i've copied from my old system to the new one
are not behaving correctly.

if i'm trying to double-click open text-file, i get dialog:

Why it doesn't open just by double-click? I need to press Display button.

New files i generate through Pluma are working normally.

How to fix?

Your copy seems to have switched-on the x (executable) bits on your text files so now the file manager thinks it might be scripts, to be run, instead of text files, to be shown.
Do an "chmod a-x" on all files which are NOT scripts or (compiled) executables (and do NOT do this on directories, they need the x).

This often happens when you copy files from/to a FAT-type file-system, like most memory sticks use, as that fs doesn't have the concept of "executable permission" so unless you got your mount options exactly right everything comes OFF the fs with all x-bits set (so that directories and "real executables" do not loose them).

I mostly mount vfat sticks with the fmask=0133 option so that NO files at all will be executable (but then - of course - would have to restore the x-bits on file I copied that do need them).
But on FAT and (I believe) NTFS executable bits is a all or nothing thing, either all files are executable or none at all.

My first guess, and it is a guess because you have not included NEARLY enough information, is that in copying the files from the older system you changed the permissions. The ownership may also need to be adjusted.

Check the permissions on the files, and turn off the x=executable flag(s). With the exception of shell command (batch) files, text files should NOT be executable.

Ok, thanks for both of you, i think dilemma solved.

Sorry, if provided information was lacking, not very
advanced user yet!